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Phys 102 – Lecture 26
The quantum numbers and spin
1
Recall: the Bohr model
Only orbits that fit n e– wavelengths are allowed
SUCCESSES
Correct energy quantization & atomic spectra
me k 2e4 1
En  
 2
2
2
n
+
n  1, 2,3,...
FAILURES
Radius & momentum quantization violates
Heisenberg Uncertainty Principle
e– wave
n2 2
2
rn 

n
a0
2
me ke
r  pr 
2
Electron orbits cannot have zero L
Ln  n
Orbits can hold any number of electrons
Phys. 102, Lecture 26, Slide 2
Quantum Mechanical Atom
Schrödinger’s equation determines e– “wavefunction”
2

ke2 
2
 

 ψ (r , θ , φ)  Eψ (r , θ , φ) 
r 
 2me
“SHELL”
“Principal Quantum Number”
me k 2e4 1
En  
2 2 n2
(  1)
n = 1, 2, 3, …
s, p, d, f “SUBSHELL”
ℓ = 0, 1, 2, 3 …, n–1
Magnitude of angular momentum
“Magnetic Quantum Number”
Lz  m
,m
Energy
“Orbital Quantum Number”
L
ψn,
3 quantum numbers
determine e– state
mℓ = –ℓ, …–1, 0, +1…, ℓ
Orientation of angular momentum
Phys. 102, Lecture 26, Slide 3
ACT: CheckPoint 3.1 & more
For which state is the angular momentum required to be 0?
A. n = 3
B. n = 2
C. n = 1
How many values for mℓ are possible for the f subshell (ℓ = 3)?
A. 3
B. 5
C. 7
Phys. 102, Lecture 26, Slide 4
Hydrogen electron orbitals
ψn,
2
,m
Shell
 probability
(n, ℓ, mℓ)
(2, 0, 0)
(2, 1, 0)
(2, 1, 1)
(3, 0, 0)
(3, 1, 0)
(3, 1, 1)
(3, 2, 0)
(3, 2, 1)
(3, 2, 2)
(4, 0, 0)
(4, 1, 0)
(4, 1, 1)
(4, 2, 0)
(4, 2, 1)
(4, 2, 2)
(5, 0, 0)
(4, 3, 0)
(4, 3, 1)
(4, 3, 2)
(4, 3, 3)
(15, 4, 0)
Subshell
Phys. 102, Lecture 26, Slide 5
CheckPoint 2: orbitals
Orbitals represent probability of electron
being at particular location
1s (ℓ = 0)
r
2p (ℓ = 1)
r
2s (ℓ = 0)
r
3s (ℓ = 0)
r
Phys. 102, Lecture 26, Slide 6
Angular momentum
What do the quantum numbers ℓ and mℓ represent?
Magnitude of angular momentum vector quantized
+
–
Classical orbit
picture
(  1)
 0,1, 2
n 1
Only one component of L quantized
Lz  m
e–
L
Lz  
1
L L
r
Lz  0
Lz  
m  ,
 1,0,1,
Other components Lx, Ly are not quantized
Lz  2
L 2
Lz  
2
L 6
Lz  0
Lz  
Lz  2
Phys. 102, Lecture 26, Slide 7
Orbital magnetic dipole
Electron orbit is a current loop and a magnetic dipole
μe
μe  IA  
r
+
–
L
μe  
e–
e
L
2me
e
L
2me
Recall Lect. 12
Dipole moment is quantized
What happens in a B field?
B
θ
z
e–
–
+
μe
r
U   μe B cos θ 
e
Bm
2me
Recall Lect. 11
Orbitals with different L have different
quantized energies in a B field
e
eV “Bohr magneton”
μB 
 5.8 105
2me
T
Phys. 102, Lecture 26, Slide 8
ACT: Hydrogen atom dipole
What is the magnetic dipole moment of hydrogen in its
ground state due to the orbital motion of electrons?
μe  
e
L
2me
e
A. μH  
2me
B. μH  0
e
C. μH  
2me
Phys. 102, Lecture 26, Slide 9
Calculation: Zeeman effect
B
Calculate the effect of a 1 T B field on
the energy of the 2p (n = 2, ℓ = 1) level
μe
m  1
z
+
Etot  En2  μe B cos θ
e
 En  2 
Bm For ℓ = 1,
mℓ = –1, 0, +1
2me
B0
e– –
Energy level splits into 3, with energy splitting
B0
m  1
m 0
m  1
e B
E 
2me
Phys. 102, Lecture 26, Slide 10
ACT: Atomic dipole
The H α spectral line is due to e–
transition between the n = 3, ℓ = 2
and the n = 2, ℓ = 1 subshells.
How many levels should the n = 3, ℓ = 2 state split into in a B
field?
E
A. 1
B. 3
C. 5
n=3
n=2
ℓ=2
ℓ=1
Phys. 102, Lecture 26, Slide 11
Intrinsic angular momentum
A beam of H atoms in ground state passes through a B field
n = 1, so ℓ = 0 and expect
NO effect from B field
Atom with ℓ = 0
Instead, observe beam
split in two!
Since we expect 2ℓ + 1 values
for magnetic dipole moment,
e– must have intrinsic angular
momentum ℓ = ½.
“Spin” s
“Stern-Gerlach experiment”
Phys. 102, Lecture 26, Slide 12
Spin angular momentum
Electrons have an intrinsic angular momentum called “spin”
S  S  s( s  1)
S z  ms
with s = ½
ms   12 ,  12
Sz  
Sz  
S
2
3
2
2
Spin also generates magnetic dipole moment
S
e
μs  
gS
2me
B
S
Spin DOWN (– ½)
B0
Spin UP (+½)
ms   12
ms   12
U   μs B cos θ 
with g ≈ 2
ge
Bms
2me
Phys. 102, Lecture 26, Slide 13
Magnetic resonance
e– in B field absorbs photon with energy equal to splitting
of energy levels
Absorption
ms   12
hf  E
“Electron spin resonance”
Typically microwave EM wave
ms   12
Protons & neutrons also have spin ½
μ prot
e

g p S  μs
2m p
since m p  me
“Nuclear magnetic resonance”
Phys. 102, Lecture 26, Slide 14
Quantum number summary
“Principal Quantum Number”,
me k 2e4 1
En  
2 2 n2
n = 1, 2, 3, …
Energy
“Orbital Quantum Number”, ℓ = 0, 1, 2, …, n–1
L
(  1)
Magnitude of angular momentum
“Magnetic Quantum Number”, ml = –ℓ, … –1, 0, +1 …, ℓ
Lz  m
Orientation of angular momentum
“Spin Quantum Number”, ms = –½, +½
S z  ms
Orientation of spin
Electronic states
Pauli Exclusion Principle: no two e– can have the same set of
quantum numbers
E
s (ℓ = 0)
+½ –½
n=2 –
–
p (ℓ = 1)
+½ –½
–
–
mℓ = –1
mS = +½
n=1
–
+½ –½
–
–
mℓ = 0
+½ –½
–
–
mℓ = +1
mS = –½
–
Phys. 102, Lecture 26, Slide 16
The Periodic Table
Pauli exclusion & energies determine sequence
s (ℓ = 0)
Also s
p (ℓ = 1)
n=1
2
d (ℓ = 2)
3
4
5
6
7
f (ℓ = 3)
Phys. 102, Lecture 26, Slide 17
CheckPoint 3.2
How many electrons can there be in a 5g (n = 5, ℓ = 4) subshell of an atom?
Phys. 102, Lecture 26, Slide 18
ACT: Quantum numbers
How many total electron states exist with n = 2?
A. 2
B. 4
C. 8
Phys. 102, Lecture 26, Slide 19
ACT: Magnetic elements
Where would you expect
the most magnetic
elements to be in the
periodic table?
A. Alkali metals (s, ℓ = 1)
B. Noble gases (p, ℓ = 2)
C. Rare earth metals (f, ℓ = 4)
Phys. 102, Lecture 26, Slide 20
Summary of today’s lecture
• Quantum numbers
Principal quantum number En   Z 2 n2 13.6eV
Orbital quantum number L  (  1) ,  0,1, n 1
Magnetic quantum number Lz  mz , m   , , 0,
• Spin angular momentum
e– has intrinsic angular momentum Sz  ms
ms   12 , 12
• Magnetic properties
Orbital & spin angular momentum generate magnetic
dipole moment
• Pauli Exclusion Principle
No two e– can have the same quantum numbers
Phys. 102, Lecture 25, Slide 21